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A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF Cn

  • Matsugu, Yasuo (Department of Mathematical Sciences Faculty of Science Shinshu University) ;
  • Miyazawa, Jun (Department of Mathematical Sciences Faculty of Science Shinshu University) ;
  • Ueki, Sei-Ichiro (Department of Mathematical Sciences Faculty of Science Shinshu University)
  • Published : 2002.09.01

Abstract

Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $\alpha$ > -1, define $dv_\alpha$(z) = $c_\alpha$$(1-\midz\mid^2)^{\alpha}$dν(z), z $\in$ B. Here $c_\alpha$ is a positive constant such that $v_\alpha$(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For $p\geq1$, define the Bergman-Privalov space $(AN)^{p}(v_\alpha)$ by $(AN)^{p}(v_\alpha)$ = ${f\inH(B)$ : $\int_B{log(1+\midf\mid)}^pdv_\alpha\;<\;\infty}$ In this paper we prove that a function $f\inH(B)$ is in $(AN)^{p}$$(v_\alpha)$ if and only if $(1+\midf\mid)^{-2}{log(1+\midf\mid)}^{p-2}\mid\nablaf\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case 1<p<$\infty$, or $(1+\midf\mid)^{-2}\midf\mid^{-1}\mid{\nabla}f\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case p = 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

Keywords

Bergman-Privalov spaces;Privalov spaces;Bergman spaces;Riesz measure;Hardy-Orlicz spaces

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Cited by

  1. Characterizations of Hardy-Orlicz and Bergman-Orlicz spaces vol.141, pp.5, 2007, https://doi.org/10.1007/s10958-007-0059-8